Scott_Aaronson2 comments on Spooky Action at a Distance: The No-Communication Theorem - Less Wrong
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Eliezer, I know your feelings about density matrices, but this is exactly the sort of thing they were designed for. Let ρAB be the joint quantum state of two systems A and B, and let UA be a unitary operation that acts only on the A subsystem. Then the fact that UA is trace-preserving implies that TrA[UA ρAB UA*] = ρB, in other words UA has no effect whatsoever on the quantum state at B. Intuitively, applying UA to the joint density matrix ρAB can only scramble around matrix entries within each "block" of constant B-value. Since UA is unitary, the trace of each of these blocks remains unchanged, so each entry (ρB)ij of the local density matrix at B (obtained by tracing over a block) also remains unchanged. Since all we needed about UA was that it was trace-preserving, this can readily be generalized from unitaries to arbitrary quantum operations including measurements. There, we just proved the no-communication theorem, without getting our hands dirty with a single concrete example! :-)